Received: 19 September 2006 / Accepted: 21 October 2006 / Published online: 2 December 2006 © Springer-Verlag 2006
Abstract In this paper, we present a continuous att-ractor network model that we hypothesize will give some suggestion of the mechanisms underlying several neural processes such as velocity tuning to visual stim-ulus, sensory discrimination, sensorimotor transforma-tions, motor control, motor imagery, and imitation. All of these processes share the fundamental characteristic of having to deal with the dynamic integration of motor and sensory variables in order to achieve accurate sen-sory prediction and/or discrimination. Such principles have already been described in the literature by other high-level modeling studies (Decety and Sommerville in Trends Cogn Sci 7:527–533, 2003; Oztop et al. in Neu-ral Netw 19(3):254–271, 2006; Wolpert et al. in Philos Trans R Soc 358:593–602, 2003). With respect to these studies, our work is more concerned with biologically plausible neural dynamics at a population level. Indeed, we show that a relatively simple extension of the classi-cal neural field models can endow these networks with additional dynamic properties for updating their inter-nal representation using exterinter-nal commands. Moreover, an analysis of the interactions between our model and external inputs also shows interesting properties, which we argue are relevant for a better understanding of the neural processes of the brain.
E. L. Sauser (
B
)· A. G. BillardLearning Algorithms and Systems Laboratory (LASA), Ecole Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
e-mail: eric.sauser@a3.epfl.ch
1 Introduction
For several decades, a large number of neural network architectures have been studied in order to understand the computational properties of the brain and its under-lying neural mechanisms. A very influential series of models that have been proposed are known as contin-uous attractor neural networks or neural fields (Amari 1977;Wilson and Cowan 1973). Indeed, in addition to their biologically plausible structural relationship with real cortical neural ensembles, their numerous compu-tational properties make them very attractive to the computational neuroscience community. For instance, these models were applied to research areas related to visual processing (Ben-Yishai et al. 1995; Giese 2000; Mineiro and Zipser 1998), visual attention (Rougier 2006), spatial navigation (Xie et al. 2002;Zhang 1996), decision making (Erlhagen and Schöner 2002; Sauser and Billard 2006;Schöner 2002), sensorimotor transfor-mations (Burnod et al. 1999; Salinas and Thier 2000; Sauser and Billard 2005), stimulus binding (Wersing et al. 2001), and parameter estimation (Deneve et al. 1999). These networks are primarily based on a center-surround recurrent connectivity, where neurons sharing similar firing properties cooperate by exciting them-selves and where neurons distant in their preferential tuning inhibit each other. This interneuron relationship is the basis of their fundamental ability to allow one sin-gle localized activity packet to emerge from this neural implementation of a winner-take-all operation.
The first point we would like to address here is con-cerned with the dynamics of stimulus–network interac-tions and, more precisely, with the temporal variation of a stimulus input in neural space. Indeed, apart from purely abstract theoretical works and a few applied to
the modeling of biological systems (Giese 2000;Mineiro and Zipser 1998;Xie et al. 2002;Zhang 1996), the use of such networks in practical neurobiological modeling assumes quasistatic neural dynamics. By quasistatic we mean that the time scale of external input changes in neural space is much larger than that of the network. Since the dynamics of neural fields are, by definition, strongly influenced by the recurrent connectivity, the reaction time to a changing stimulus is thus higher than that of single neurons (Panzeri et al. 2001). Hence, in quasistatic network dynamics, the time scale of the input updating is set to a large value. When comparing these network implementations with real biological systems, this issue is not really a problem since their performance is considered in relatively slow tasks. However, when dealing, for example, with precise and fast movements like catching a ball or smooth eye pursuit, the inter-nal representations should be updated very quickly, or even in advance, in order to predict accurately the out-come of self-generated movements (Miall and Wolpert 1996;Wolpert and Kawato 1998). An influential theory related to motor control suggests that the brain may use forward models in order to better control movements, arguing that, in humans, a closed-loop control system alone would be relatively inaccurate, given the long time needed for sending motor commands and receiving the resulting sensory feedback (Miall and Reckess 2002; Miall and Wolpert 1996;Wolpert and Kawato 1998). For instance, in an eye-tracking of a self-moved target exper-iment, it has been shown that, when the subjects actively move a target, the presence of the movement sensory feedback is not necessary to achieve almost zero latency, in contrast to the motor efference copies (Vercher et al. 1991). Furthermore, when considering neurophysiolog-ical data, predictive neural responses were found in the monkey visual, parietal, and frontal cortices and in the cerebellum (Nakamura and Colby 2002;Roitman et al. 2005;Unema and Goldberg 1997), highly suggesting the presence of forward control.
A second, more practical, motivation concerns the neural mechanisms of self-awareness and recognition. A current theoretical framework related to this research topic mainly considers two forms of cues that may be at the origin of such a human capacity (Decety and Sommerville 2003;Haggard and Clarke 2003;Jeannerod 2003). When one has to recognize one’s own limb, body cues such as the spatial and visual attributes of the limb are primarily used. Since it is not within the scope of this paper to address the problem of self-limb recog-nition from these visual attributes, we are rather more interested in the second form of cues. Indeed, when the former attributes are ambiguous, it has been shown that one relies more on action or movement cues, e.g., the
time course of the movement velocity and its accel-eration (Jeannerod 2003). A plausible mechanism has been suggested whereby an internal prediction of the consequences of a motor act is compared with the real sensory outcome. Then, depending on their similarity or discrepancy, the brain can determine the ownership of the observed movement (Decety and Sommerville 2003). In addition, another problem arises when con-sidering the possible neural substrates concerned with the perception of self action and that of others. Indeed, the current body of evidence suggests that a common neural substrate devoted to the recognition and produc-tion of movements exists in both humans and monkeys (Iacoboni et al. 1999;Rizzolatti et al. 2001). A behavioral correlate of such a discovery, described in several psy-chophysics experiments (Chaminade et al. 2005;Kilner et al. 2003), is that observing movements of others influ-ences the quality of one’s own performance. The obser-vation of such an interference effect, while supporting the view of a common pathway for the transfer of visuo-motor information, calls for an explanation as to how the same substrate can both integrate multisensory infor-mation and determine the ownership of the observed movement.
Despite the apparent differences between the two previously introduced topics, they both share a com-mon fundamental property: their need for a predictive forward model, which would allow, respectively, (a) an almost instantaneous updating of the internal represen-tation of the sensory states and (b) the compurepresen-tation of sensory predictions to be compared with movement out-come. Indeed, the timing of neural updating for internal sensory information is crucial in motor control for accu-rate movement generation (Miall and Reckess 2002; Vercher and Gauthier 1988). Similarly, a short time between movement execution and the perception of sen-sory feedback is also crucial for perceiving the agency of an action (Haggard and Clarke 2003). Our interest here is effectively to show how a neural field, a neu-ral substrate for representing information, can integrate the efferent commands from a forward model in order to update its internal representation. As internal rep-resentations, we restrict our modeling by considering neural ensembles encoding simultaneously a variable value and its first-order time derivative. For instance, the position and the velocity of either a visual stimu-lus in retinal space or the hand location in cartesian or joint space may be considered. Indeed, real popula-tions of neurons with such a tuning property have been found in several brain areas, such as the motor, parietal, visual, and temporal cortices and in the cerebellum (Ben Hamed et al. 2003;Hubel and Wiesel 1977;Jellema et al. 2004;Kettner et al. 1988;Roitman et al. 2005). The
describe and analyze properties and applications result-ing from the proposed integration mechanism, such as (a) dynamic or input driven velocity tuning, (b) instan-taneous and predictive-like information transfer, and (c) abilities for stimuli discrimination based on their dynamic properties.
This paper is organized as follows. Section2 intro-duces the model core architecture and then describes the mechanisms underlying the integration of velocity commands. Next, the form of stimulus encoding and the synaptic projections for transferring information across neural populations are defined. Note that detailed math-ematical calculations were left in the Appendix. Fur-ther, in Sect.3, we show some analysis of the network dynamic properties with and without external stimulus inputs, such as velocity tuning. We also address the tim-ing of information transfer across neural populations and then tackle the problem of stimulus discrimination. Moreover, we also compare our model dynamics with that of a more commonly used neural field architec-ture and then show its weaknesses. Finally, we discuss in Sect.4 the relationship between our model properties and biological data and present some implications and predictions from our work regarding several of the brain neural mechanisms.
2 Model
We consider a continuous attractor neural network com-posed of a set of neurons preferentially tuned to a pri-mary variable r and a secondary variable s following a uniform distribution such that r ∈ Dr and s ∈ Ds. As mentioned previously, these neural spaces are assumed to encode, respectively, a variable value and its varia-tion in time. For example, if we consider that a stimulus location is encoded in the spaceDr, a neuron tuned to a specific s will fire preferentially when the stimulus is moving in the direction given by that s. The present hypothesis regarding a combined preferential tuning to both a variable value and its direction of variation is mainly motivated by several neurophysiological stud-ies showing neurons having similar firing propertstud-ies (Fu et al. 1997;Jellema et al. 2004;Kettner et al. 1988;
Roit-man et al. 2005). In addition to the computational power that such a neural representation may provide to other connected brain networks, we argue and will show that it may also help in updating itself.
Further, the cases of both a unidimensional and a two-dimensional (2D) attractor are addressed (Table1). To avoid boundary effects, periodic spaces are assumed such that these domains form, respectively, a ring and its 2D analog, a torus. Despite the discrete nature of s in the case of the ring attractor, the neural ensemble follows continuous attractor network dynamics where the time evolution of the neurons’ membrane potential u(r, s, t) satisfies
τ ˙u(r, s, t) = −u(r, s, t) + h(s, t) + x(r, s, t) +‹ W(r − r) − λ∇W(r − r) · s
× fu(r, s, t)drds. (1) The transfer function f(u) is the linear threshold func-tion max(0, u), and τ ∈ R∗+is the time constant of the neurons. The network receives external inputs that were separated into two distinct forms. x(r, s, t) is defined as the stimulus input, whereas h(s, t) is the background input. Note that h(s, t) is, by definition, homogeneous across the subpopulations of neurons sharing the same preferential tuning to the variable s. Henceforth, these subpopulations will be designated as sublayers.
The network is fully connected by means of a set of synaptic weights composed of two parts: a center-sur-round Gaussian-like,1 translation-invariant, and sym-metric component W(r − r) and an asymmetric term −λ∇W(r − r) · s, where∇ corresponds to the gradient operator along r.λ ∈ R∗+is a constant scaling factor. The first term is a convolution kernel that links the neurons along the neural fundamental variable r, while the sec-ond term links them along s. These convolution weights and the network architecture are illustrated in Figs. 1 and2. As will be described in Sect.2.1, this connectivity is the basis of the system’s ability to integrate velocity commands.
1 The exact shape of W is not crucial for our argument, as long
as it allows the network to sustain an activity packet on its neural surface (Amari 1977). Nevertheless, in AppendixC, we define the exact shape of W that we will use further in our experiments.
W (r− r ) r r r r λ∇W(r − r ) · s W (r− r ) −λ∇W(r − r ) · s f u(r , s , t) f u(r, s, t) f u(r , s , t) f u(r, s, t) s = 1+ s =_1 W (r−r )−λ∇W(r−r ) · s f u(r, s, t) f u(r , s , t) s s
Fig. 1 Illustration of model architecture and weight kernels for both the ring (top) and the torus attractor spaces (bottom). Each sublayer encoding a variable in the neural space defined on r has a preferred movement direction s that is the result of the asym-metric self-connectivity. This weight kernel is shown for different values of s. It is superimposed on the arrows denoting the synaptic projections across the sublayers
As illustrated in Fig.2, this type of neural dynamics are known to form an activity packet or attractor bump on the surface of the neural field. It is suggested that this class of networks conveys information by such a compact blob of excitation. A typical read-out mecha-nism, the population vector p(t), has been suggested to measure the macroscopic effect of the joint activities of large sets of neurons (Georgopoulos 1996). It consists of a weighted summation of the firing activity of each neu-ron with its preferred direction. The estimate p(t) ∈ Dr of the variable value encoded along r is given by
p(t) = ‚
fu(r, s, t)r dr ds ‚
fu(r, s, t)dr ds . (2)
An estimate of the encoded variable in Ds might be envisaged similarly. However, since that space is an extension of the primary spaceDr used for integration purposes, its explicit definition is not necessary. The next section addresses the use of that dimension for updating the network neural representation using external com-mands provided through the background input.
2.1 Forward model commands and intrinsic dynamics
In this section, we are interested in how forward model commands, by acting on the background input, may drive the network internal dynamics so that the neural representation may be updated accordingly. Afterwards, we also determine the boundaries in which this velocity integration remains valid. Let us start by defining how the population vector p(t) encoded by the neural repre-sentation may be varied by an external command v(t) such that
˙p(t) = v(t). (3)
Furthermore, we consider here that the network is already representing and sustaining information and that the stimulus input is absent, i.e., x(r, s, t) = 0. The background input h(s, t) is then defined as
h(s, t) = h0[1 + ˆh0(t) · s], (4)
where h0> 0 is a constant excitation and ˆh0is a direc-tional input component that can break the network sym-metry when different from zero (Fig.3a). Indeed, when referring to Eq. (1), it can be noticed that a strictly homo-geneous input ( ˆh0 = 0) leads to a zero value of the integral along s of the asymmetric weights convolution, i.e.,
‹
λ∇W(r − r) · sfu(r, s, t)drds= 0 . (5) By symmetry, the system thus settles into a constant state along s. In this case, this network is equivalent to a typi-cal continuous attractor network without an asymmetric connectivity, which is known to exhibit marginally stable bump solutions.
Further, an input ˆh0 = 0 breaks this symmetry. It consequently favors a higher excitation of the network sublayers with a direction preference s close to that given by ˆh0. Now, if we first consider a single sublayer, it has been shown that, as an attractor state, it develops a trav-eling activity bump symmetric in shape and with con-stant velocity. That speed depends strictly on the ratio between the strength of the asymmetric weights and the neuron’s time constant and is given byλ/τ (Zhang 1996). Coming back, then, to the full architecture, the coupling across the sublayers introduces a competitive
Fig. 2 Illustration of model architecture. Each sublayer encoding a variable in the neural space defined on r∈Drhas a preferred
movement direction s ∈ Ds that is the result of the
asymmet-ric self-connectivity. a The center arrows indicate the preferred movement direction of the sublayers. If considered alone, each
of them would display a traveling activity blob in its preferred movement direction, as shown by the trace of the activity blobs on the neural surface. b Each sublayer projects its activity onto all sublayers using the same weight profile
interaction between those having an opposite direction preference. This produces a push–pull mechanism regu-lated by the respective balance of excitation among the sublayers. In our model, this effect is controlled by the strength of the asymmetry of the background input ˆh0. Therefore, by balancing this input factor, one may drive the network representation p(t) in direction and veloc-ity. The detailed calculations given in AppendixAshow that near the equilibrium, the relationship between p(t) and ˆh0(t) can be approximated linearly such that
˙p(t) ≈ λτ γ ˆh0(t), (6)
where ˆh0(t) is small. γ is the slope of the linear approx-imation that depends on the recurrent weight parame-ters.2Since this updating of the neural representation is performed by modulating the respective influence of opposite sublayers, the maximal velocity that our model may attain is constrained by that of a single sublayer, i.e., λ/τ. Thus, the network velocity is bounded such that ˙p(t) ≈ λτ min γ ˆh0(t), ˆh0(t) ˆh0(t) . (7)
Then, rewriting Eq. (4) using Eqs. (3) and (7) gives
h(s, t) = h0 1+ τ λγ v(t) · s v(t) λ τ , (8) which corresponds to the background homogeneous input to be applied to the network so that it can drive its internal representation with speed v(t). Like a related model (Xie et al. 2002), this integration mechanism makes it possible for forward commands to allowthe
2 Except for rare special cases,γ must be found numerically.
network to predict its next state p(t) according to its supposed variation in time. As a consequence, the sta-tic marginal attractor states of the model become linear trajectories with constant velocity in the neural space defined on Dr. Therefore, to contrast the effect of a stimulus input on the network, our network is said to possess an intrinsic velocity v(t) < λ/τ determined by its background input, which is defined as the velocity of the trajectories of its attractors.
2.2 Stimulus encoding
In this section we describe how a stimulus input should be defined in order to drive the network dynamics toward that of the input. A nonlinear form is first defined, since its derivation from the background input is the most straightforward. We will then slightly mod-ify this description by approximating it with a linear form more suitable for transferring information across multiple neural populations. Indeed, this linear input form will further help us derive appropriate synaptic projections. As a consequence, within a large network, instances of our model may easily transfer their posi-tional and velocity-dependent information across each other. So, let us introduce the first form of stimulus input. We consider a stimulus located at r0(t) in the neu-ral space Dr moving in phase with the network intrin-sic velocity, i.e., ˙r0(t) = v(t). Its input to the network x(r, s, t), shown in Fig.3b, is given by
x(r, s, t) = h1 Gr− r0(t)− τ ˙r0(t) · ∇Gr− r0(t) 1+ τ λγ ˙r0(t) · s ˙r0(t) λ τ , (9)
where h1> 0 corresponds to the stimulus amplitude and G to the fundamental shape of the input. We assume this shape to be Gaussian-like and centered on the stimulus location r0(t) (see a precise definition in AppendixC). The second term of the first factor compensates for the neuron’s integration time so that the network represen-tation of the input shape is unaffected by stimulus move-ment. It stays symmetric, as if it was static in the neural medium. Finally, comparing the equation of the stimu-lus input with that of the background input [Eqs. (9) and (8)], it can be noticed that the asymmetric term in the second factor is responsible for driving the push–pull mechanism of the neural field. Through this modula-tory excitation, the asymmetric recurrent connectivity will help the neural field tracking the stimulus. Since the asymmetry of the stimulus input is driven by its speed, we introduce here a directional input factor ˆh1that has a role similar to that of the background input. It is given by
ˆh1(t) = λγτ ˙r0(t). (10)
This relationship relates to the amount of strength of the input asymmetry that is necessary for the network to follow its speed. Rewriting Eq. (9) then gives
x(r, s, t) = h1 Gr− r0(t)− λγ ˆh1(t) · ∇Gr− r0(t) 1+ ˆh1(t) · s . (11)
Since the asymmetric factors of both the stimulus and the background input are proportional, respectively, to the stimulus and intrinsic network velocity, the use of either notation will be considered equivalent in the fur-ther analysis of our experiments. Again, the reader inter-ested in a more detailed description of this mathematical development is encouraged to refer to AppendixB.1.
2.2.1 Linear encoding
As mentioned earlier, an alternative stimulus input form can be defined so as to avoid the nonlinear multiplica-tive factor found in its previous definition [Eq. (9)]. This multiplicative factor is needed to drive the internal net-work dynamics toward that of the input. In order to replace it, the idea is to apply the same principle as that described in the case of the background input. A con-stant term, homogeneous along the stimulus spaceDr and with a velocity-dependent asymmetry along s, can be used. Detailed calculations given in Appendix B.2
result in the following input form:
x(r, s, t) = h1 Gr− r0(t)− τ ˙r0(t) · ∇Gr− r0(t) +η ˆh1· s , (12)
whereη is a constant that depends on both the network recurrent weights W and on the input shape G. This form of linear encoding is particularly interesting for trans-mitting information across populations. Indeed, as will be described next, our model may transfer both its posi-tional and velocity-related information to another field by means of strictly linear synaptic projections chosen appropriately.
2.3 Information transmission across neural fields
In large-scale neural networks, it may be envisaged that several instances of our model may communicate and transfer information. The synaptic projections Wabfrom one neural field A to another B are defined so that the input xb(r, s, t) of population B is given by
xb(r, s, t) = ‹ Wab(r, s, r, s)f ua(r, s, t) drds. (13)
In this section, our aim is to allow these synaptic projec-tions to be equivalent to the stimulus input form given by Eq.12. Consequently, the information conveyed by the source population A would be completely transferred such that pb(t) = pa(t) and ˙pb(t) = ˙pa(t). Usually, in clas-sical neural field implementation, information is trans-mitted using a weight kernel having a symmetric and center-surround shape. However, as shown previously, the ability for velocity integration is the result of an asymmetric coupling across the network sublayers. Thus, we here use a similar technique to that described in the context of whole network dynamics [Eq. (1)]. It can be shown (AppendixB.3) that the following weights
Wab(r, s, r, s)=
Wt(r−r)−λ∇Wt(r−r) · s+μabs· s (14)
produce the desired effect. Wtis a symmetric and cen-ter-surround convolution kernel strictly defined on the neural space Dr. The last term, μabs· s, where μab is a constant depending on the recurrent weights and on the input shape, is the analog to that of Eq. (12). The input source may hence drive the network according to its own dynamics.
Fig. 3 Examples of external inputs where the same representa-tion as in Fig.2is used. The color code on the right indicates the corresponding preferred movement direction of each sublay-er. a Background input h(s, t) [Eq. (4)]. b stimulus input x(r, s, t)
[Eq. (9)]. c Linear form of stimulus input [Eq. (12)]. The arrow in the middle of the figures denotes the intrinsic velocity of the network resulting when the corresponding input is applied
3 Experiments and results
In this section, we present several experiments that were performed using our model. Sections3.1and3.2show numerical simulations aimed at illustrating the relevance of our mathematical developments. Our model’s abil-ity to integrate velocabil-ity commands and its response to both types of stimulus inputs and synaptic projections across multiple instances of our networks are described. We further investigate, in Sect.3.3, how our model may reproduce neurophysiological data concerning the pref-erential tuning to stimulus velocity of several groups of neurons in the visual cortex (Orban et al. 1986). Finally, we report two additional simulation results addressing mechanisms that we suggest to be respectively related to motor imagery and sensory discrimination. The for-mer experiment describes how the excitation level of the network may liberate it from its sensory influences. The latter considers a neural population that receives two contradictory and ambiguous sensory feedback inputs and that should select the input corresponding best to its own internal dynamics.
3.1 Dynamic velocity integration
In the hippocampal formation of the rat, it has been shown that the so-called head-direction cells code for the current heading direction of the animal. It has also been observed that this group of neurons displays a fast updat-ing of its representation accordupdat-ing to motor efference copies or vestibular inputs (Sharp et al. 2001). Moreover, in other brain regions, neural representations encoding for different sensory states have also been shown to exhibit a similar updating (Roitman et al. 2005;Schwartz and Moran 1999).
Here, we show that our model may also update its internal representation according to the commands v(t) given through the background input h(s, t). The simula-tion parameters that were used in each of the following experiments are summarized in AppendixC. Figure4 shows the measured velocity response ˙p(t) of the neu-ral field as a function ofˆh0. Each point on the graph was obtained in a different trial with a different value for ˆh0. The approximation, given by Eq. (7), is dis-played in the same graph. Note that, in the ring attractor case, for a sufficiently strong asymmetric input, the net-work reaches the maximum velocityλ/τ. However, in the torus case, it can be seen that the system only tends toward that maximum asymptotically. Indeed, in the for-mer case, the stronger sublayer is capable of completely inhibiting its opposite and hence driving the whole net-work alone. The continuous nature of the direction pref-erence in the second case forbids a single sublayer to win against all the others. Nevertheless, for a relatively small asymmetric component ˆh0, the theoretical approxima-tion fits the simulaapproxima-tion results well. In Figs. 5 and 6, some trajectories of the command v are compared to the velocity response ˙p(t) of the network.
3.2 Information transfer
The purpose of this section is to show how our net-work can integrate the representation given by a stim-ulus input and by projections from another network. It was mentioned previously that a population of neurons may update its encoded variable through the integration of velocity commands. Similarly, a stimulus input should also be able to do so. In what follows, our model will be shown to capture such an effect. In addition, we will also compare its performance with that of a more commonly used single-layered attractor network. Recall that the
0 Velocity λ/τ 0.5 0 λ/τ 0.5 h0 ^ p(t)
Fig. 4 Network velocity response (dotted line) to different asym-metric background input drives ˆh0, as displayed by a ring (left)
and torus attractor (right). The straight line corresponds to the approximation of the velocity response [Eq. (4)]
Velocity
Time
λ/τ 0
10τ τ
Fig. 5 Velocity response˙p(t) of a ring network to external com-mands v(t) provided through background input [Eq. (8)]. Dotted and straight lines correspond, respectively, to the desired velocity command v(t) and the network velocity response ˙p(t). As can be seen when a high-velocity command is given, the network satu-rates at its maximum velocity integration boundary given byλ/τ
Velocity along e1 Time λ/τ 0 Time λ/τ 0 Velocity along e2 10τ τ
Fig. 6 Velocity response˙p(t) of a torus network to external com-mands v(t). Data are shown relative to the two principal axes of the network representation spaceDrgiven by the canonical base
{e1= (1, 0), e2= (0, 1)}. The same notation as in Fig.5is used
latter neural dynamics are given by
τ ˙u(r, t) = −u(r, t)+x(r, t)+h(t) + ˛
W(r−r) f (u(r, t))dr. (15)
When compared to Eq. (1), this corresponds to a net-work with a reduced dimensionality. In this case, the stimulus and the background inputs are given, respec-tively, by x(r, t) = h1G(r − r0(t)) and h(t) = h0.
3.2.1 Transfer from stimulus to network representation To illustrate the validity of our mathematical develop-ment concerning stimulus integration, we applied sepa-rately to the network the nonlinear and the linear forms of the stimulus input [Eqs. (9) and (12), respectively]. The background input amplitude h0was set to zero. For different stimulus speeds, we measured the spatial lag between the effective stimulus spatial location r0(tn) and the neural population vector p(tn) at a given time tn. The results are plotted in Fig.7a and b. These figures show that, under both conditions, our model outperforms the simpler model. Indeed, below the maximum integration velocity given byλ/τ, the lag stays close to zero, while an almost linear velocity-dependent lag is observed for the other model.
3.2.2 Transfer across network representation
Next, we performed similar simulations while consider-ing the transfer of information across the representa-tions of two interconnected neural fields using Eqs. (13) and (14). The projection weights kernel Wtis defined in AppendixC. Figure7c shows the resulting lag measured across the neural populations. As expected, the lag effec-tively stays close to zero below the velocity integration boundaryλ/τ, which corresponds to a property similar to that mentioned in the case of a stimulus input.
3.3 Dynamic velocity tuning
In the context of the visual system, neurons have been found to be preferentially tuned to both a stimulus posi-tion and velocity in retinal space. This property has further been suggested to be the basis of the human ability for velocity discrimination (Cheng et al. 1994; Chey et al. 1998;Goodwin and Henry 1975; Mineiro and Zipser 1998). Our interest is to see if our model may first reproduce this experimental result and further suggest an alternative approach to this neural process. Indeed, it has already been shown that a single-layered attractor network having a fixed asymmetric recurrent connectivity can exhibit a certain tuning to input stim-ulus speed (Mineiro and Zipser 1998). However, such a model is restricted to be tuned to a single preferred velocity. It thus needs to be replicated a sufficiently large number of times with different synaptic strengths so that
0 1 0 λ/τ = 1.0 λ/τ = 0.5 λ/τ = 1.5 r0 0.5 1.5 0 1 0 r0 0.5 1.5 0 1 0 r0 0.5 1.5 λ/τ = 1.0 λ/τ = 1.5 λ/τ = 1.0 λ/τ = 1.5 0.5σ 0.25σ 0.5σ 0.25σ 0.5σ 0.25σ (a) (b) (c)
Fig. 7 Bottom Positional lag at a given time tnbetween the
net-work representation p(tn) and a a moving stimulus encoded using
the nonlinear method (Eq.9), b a moving stimulus encoded with the linear method (Eq.12), and c another network driven by a moving stimulus (Eq.13). To quantify the amount of latency between these representations, the lag is given relative to the breadth σ of the input shape G (Appendix C). Top System schematic shows which part of the network is considered in each
case. Data are shown for different values of the ratioλ/τ, which corresponds to the maximum allowable integration velocity. Note that the lag is very low below this maximal value but then grows almost linearly with the input speed. Moreover, this figure also compares our neural field model with classical continuous attrac-tor network implementation (dotted line). Note that such a model suffers from a lag that is almost linearly related to the input speed
the whole system will then exhibit a preferential tuning to a broad range of velocities (Chey et al. 1998).
By considering the changes in the network response resulting from the modulation of the stimulus speed, we show that our model can display this large range of pref-erential tuning by simply varying the asymmetry of the background input h(s, t). Indeed, by dynamically setting the intrinsic network velocity v, a stimulus with a close speed will resonate or cooperate more with the network than divergent ones. Since cooperative interactions in such an attractor network result in higher activation pat-terns than competitive ones, we assume that the mean global firing rate of all the neurons is a good measure of this resonance effect. The network response energy E(t) is defined by
E(t) = ‹
fu(r, s, t)drds . (16)
Several simulation trials were performed while the model was given various intrinsic velocities v. Then, a single stimulus moving according to a large range of speeds and directions was applied. The measured net-work energy for each trial is shown in Fig. 8. It can be seen that the network effectively responds prefer-entially to stimuli with a close velocity and that these tuning curves show a high similarity to those reported in the visual cortex (Cheng et al. 1994;Orban et al. 1986).
3.4 Stimulus to background input strength ratio
The experiments described above were mostly perfor-med while the mean background homogeneous input h0 was kept sufficiently small so that the influence of the network recurrent connectivity was relatively weak compared to the strength of the stimulus input h1. Indeed, the ratio h0/h1is of critical importance when considering the network dynamic behavior. Indeed, it determines which of these inputs is driving the network. A small value corresponds to a predominance of the stimulus input, while a larger one corresponds to a dom-inance of the network intrinsic dynamics.
This network property may allow the same neural substrate to be used for different purposes. For instance, several motor areas of the brain have been shown to be activated similarly by movement execution, imagery, and observation (Fogassi and Gallese 2002;Jeannerod and Decety 1995;Porro et al. 1996;Rizzolatti et al. 2001), processes that are not completely equivalent. During motor execution, it is necessary for the brain to keep track of the real sensory states, whereas during the other processes, it should avoid perceiving them. Consider-ing now an analogy for our model, in the former case (h1 h0), the network stays locked to the input stimu-lus. In the latter case (h0 h1), the intrinsic dynamics become sufficiently strong to free itself from the stimulus
0.2 0.4 0 1 E h0 ^ h1 ^ h1 ^ E h0 ^
Fig. 8 Top Ring network mean energy relative to the speed of the input stimulus. Each plot corresponds to a different intrinsic network speed. Notice the shift in the preferential velocity tuning
of the network with its intrinsic speed. Bottom The same results for a torus attractor. The black dot indicates the network intrinsic speed. The brighter the plot is, the higher the energy
input. Hence, the network evolves strictly according to its background input. Figure9shows, for different values of the ratio h0/h1, the velocity space in which the net-work is mostly driven by either the stimulus or the back-ground input. By mostly driven by the stimulus input we mean that the network response p(t) is strongly depen-dent on the stimulus location r0(t). Indeed, if the lag between these values is almost constant through time, it means that the network actually follows that input. In contrast, an uncorrelated difference between p(t) and r0(t) indicates that the network is only driven by its back-ground input. The data shown in Fig.9was obtained by first letting the network and the stimulus evolve for a given and sufficiently long period of time. Then, we mea-sured the lag between the stimulus location and that of the network representation. If, for a given trial, the lag was less than twice the breadth of the stimulus inputσ (AppendixC), the network was considered to be stimu-lus driven.
3.5 Sensory discrimination
The last simulation concerns sensory discrimination. Our inspiration was taken from a behavioral experi-ment addressing the recognition of a stimulus moving according to self-generated movements in the presence of one ambiguous distractor. Indeed, the neural mech-anisms related to self-recognition are supposed to be grounded in the brain by means of a forward model taking as inputs motor efference copies whose predic-tions are compared with actual sensory consequences (Decety and Sommerville 2003). A close match between the prediction and the sensory feedback is supposed to signify that the observed stimulus is controlled by the self, whereas a discrepancy would mean that it is
under an external influence. Neurophysiological data suggest that this discrimination process may be partly grounded within brain regions containing shared repre-sentations (Decety and Sommerville 2003). This would signify that a neural population receiving two ambiguous inputs should be capable of performing such a selection. Therefore, we applied this principle to our model. Two external stimulus inputs located, respectively, at rc0(t) and ri0(t) in neural space are fed to our network. Since they are supposed to be ambiguous, their corre-sponding amplitude is equivalent, i.e., hc1= hi1, and their initial location in neural space is assumed to be identi-cal. Respectively designated by the indices c and i, the compatible stimulus has a speed that is equally fed to the network through its background input, i.e., ˆhc1= ˆh0 such that ˙rc
0(t) ≈ v(t), while the incompatible stimu-lus is different. Simulation results are shown in Figs.10 and 11. Figure10illustrates the network neural activ-ity at different time steps. It can be seen that when the stimulus inputs are separated in neural space, the net-work selects the input with a speed corresponding best to its own intrinsic dynamics. Figure11shows the tem-poral dynamics of stimulus selection for different cases. Below the maximum intrinsic network speed (Fig.11a, b), the selection is successful, but above, the network may either lag behind the compatible stimulus or even select the incompatible stimulus (Fig. 11c, d). Finally, Fig.12summarizes the range of speeds where the net-work successfully discriminates the right stimulus. In the case of a right decision, the spatial lag between the pop-ulation vector p(t) and the compatible stimulus location rc
0is shown. Note that the lag is almost zero when the compatible stimulus velocity lies within the allowable range of velocity integration. Indeed, as described in Sect. 3.2.1, outside that range, the lag increases with
where the background input dominates (dashed arrows)
0 0 0.4 1.0 1 1 1.6 h1 ^ Stimulus driven Background input driven h0 h1
Fig. 10 Network response during a velocity discrimination task. Each subplot corresponds to a snapshot of the membrane potential
u(r, s, t) of selected sublayers sorted according to their preferred
movement direction. The surfaces enclosed with the white dotted
line indicate regions of neural space in which activity is above
zero, i.e., the neurons within these areas are actually firing. In the
middle of each subfigure, the external input x(r, s, t) averaged over
the preferred directions of movement s are shown. a Beginning
of discrimination task: both the compatible and the incompatible stimuli are at the same location in neural space; they are indis-tinguishable. b The stimuli, moving at different speeds, start to separate but are not distinguishable on the network representa-tion yet. c The stimuli are clearly disjointed, and, by means of its recurrent interactions, the network naturally selects the stimulus that is the most compatible with its own intrinsic velocity
the stimulus speed. Moreover, the regions of false dis-crimination are strictly located over the system inte-gration boundary. Since the network cannot accurately
update its internal representation, it naturally selects the input that is the closest to its actual intrinsic velocity. When the input with incompatible speed is closer to the
0
(a) (b) (c) (d)
Time
Position Compatible stimulus Incompatible stimulus Internal representation p(t)
r0(t)
r1(t)
Fig. 11 Illustration of network response p(t) when a stimulus compatible with the intrinsic network velocity and an incompati-ble one are applied to the inputs of the network. The positions of the compatible stimulus (light filled line), the incompatible stim-ulus (dotted line), and the network response (dark filled line) are shown over time in several situations. a–c As soon as the stimuli are sufficiently separated in neural space, the network success-fully selects the compatible stimulus. Note that the trajectory of the internal representation p(t) exhibits, initially, a deviation from
the compatible stimulus trajectory. As described in Fig.10, this is the result of the temporary overlap between the two stimulus representations on neural space; the network thus cannot discrim-inate between the two. However, as soon as the stimuli are suffi-ciently separated, the selection is performed. c Since the speed of the compatible stimulus is above the maximal network velocity, a constant lag can be observed. d In a similar out-of-bounds situa-tion, the network selects the wrong input. Indeed, the input speed is here closer to that of the network
integration boundary than that of the compatible one, it is selected.
These results confirm that our model can account for the importance of the precise timing between the pre-diction of the movement outcome and the sensory feed-back. Indeed, this allows our model to select which of the stimuli is under self-control (Haggard and Clarke 2003). Consequently, it also proposes a neural mecha-nism contributing to this cognitive function and suggests how it may be realized within a shared representation.
4 Discussion
In this paper, we have presented a continuous attrac-tor network model capable of integrating velocity com-mands for updating its internal representation. Moreover, we have also considered and defined ade-quate external input forms such that they can positively influence the network integration dynamics toward their own. Further, the analysis of the network dynamics reveals that our model should not be restricted to repre-senting the mechanism of a single brain region. Rather, its various properties suggest that its architecture might be more dispersed across brain areas, such as the cere-bellum and the motor, visual, and associative cortices.
Other models addressing the integration of velocity commands have already been described in the literature. The vast majority of them consider the rat head direc-tion system, its hippocampal places fields, and its abilities for path integration (Redish et al. 1996;Stringer et al. 2004;Xie et al. 2002;Zhang 1996). Despite the fact that the biological plausibility of their implementation is still under debate, a series of models assumes the existence of
Fig. 12 Range of compatible-incompatible stimulus velocities leading the network to a false decision (triangular black regions). Moreover, in the case of a correct decision, the figure also shows the lag measured between the location rc
0of the compatible
stim-ulus and the network response p. The lag is given relative to the breadthσ of the stimulus input. As can be seen, when the com-patible stimulus speed is over the network integration limit, an increasing lag between the input and the network response is observed
sigma-pi units, i.e., neurons performing both a sum and a product of their inputs (Redish et al. 1996;Stringer et al. 2004; Zhang 1996). Although our architecture allows for avoiding the use of such computational units, our model is nevertheless largely inspired from the compu-tational principles that were described by Zhang (1996) and may even, under certain assumptions, be reduced to his model. Indeed, by averaging over the sublayers sharing the same preferential movement direction, only
has given several new predictions and hypotheses that will be described throughout the following discussion.
4.1 Velocity tuning
Velocity and direction tuning are properties of neurons that have been found primarily in visual cortices, such as V1 and V2. They have been shown to fire preferen-tially for stimuli moving at a specific speed and direc-tion within their receptive field (Orban et al. 1986). This neurophysiological finding has been suggested to be the basis of the human ability to discriminate veloc-ity (De Bruyn and Orban 1988). Earlier work by Mineiro et al. (1998) has already demonstrated how a neural field endowed with a fixed asymmetric recurrent connectivity can exhibit such a sensitivity. However, this network is constrained to exhibit a single velocity tuning (Mineiro and Zipser 1998). Consequently, in order to be sensitive to a broader range of velocities, this network should be replicated a large number of times with different weight strengths. This approach has been followed by other models addressing the problem of stimulus velocity dis-crimination (Chey et al. 1998). In contrast, the present model proposes a mechanism relying on sublayers hav-ing opposite directional tunhav-ing, where the precise veloc-ity tuning is then driven dynamically by the background input. Indeed, velocity discrimination is a dynamic pro-cess where a stimulus is first presented to subjects so that they can internalize its velocity for further discrimi-nation against other stimuli. In addition, although it has been shown that visual areas, such as V1 and V2, possess a wide range of velocity-tuned neurons (Goodwin and Henry 1975;Orban et al. 1986), these areas are likely to be located in earlier stages of the whole visual pro-cess, as compared to area MT. This area sends direct projections to the parietal cortex, which is suggested to be the locus of some decisional processing (Wise et al. 1997). Therefore, the information carried by MT is more likely to be responsible for the discrimination process. Moreover, in this area, it has been shown that the distri-bution of velocity-tuned cells appears to be not uniform but rather centered near some extreme values (Cheng et al. 1994). These findings seem to confirm our model-ing approach. It suggests that the brain may be able to
4.2 Sensorimotor transformations and motor control
The transmission time of information across neural structures is critical for time-dependent tasks such as movement control. However, the brain is known to suf-fer from delays arising from the substrate where infor-mation is encoded, i.e., the neurons. In addition to the single neuron’s integration time constant, the high den-sity of the recurrent connectivity among cortical col-umns and across brain regions, while providing the brain with high computational power, adds an even stronger inertia to the information flow and hence increases the overall system time constant. Both neurophysiological and modeling studies have already described the con-sequences in response latency of intralayer recurrent and center-surround connectivity during simple stim-ulus-response tasks (Panzeri et al. 2001;Raiguel et al. 1999). Moreover, the recurrent connectivity has another side effect, not captured by these studies, which is the difficulty of moving from one attractor state to another. This problem has several implications regarding recently proposed neural mechanisms that may be the basis of sensorimotor transformations (Burnod et al. 1999;Deneve et al. 1999;Salinas and Thier 2000;Sauser and Billard 2005; Scherberger and Andresen 2003). Indeed, these approaches rely on populations of neu-rons encoding basis functions and grouped within gain fields, which are neural substrates connected recipro-cally and which receive changing inputs from several external sources. For example, to compute the location of a visual target in body-centered coordinates, the brain is supposed to merge information related to the target location in retinal space, to the displacement of the eyes relative to the head, and to the direction of the head in body-centered coordinates. Then, despite the inertia resulting from the recurrent connectivity, the desired information must be read out from this mixed repre-sentation with a latency as short as possible. Neverthe-less, the brain successfully performs that operation. For instance, in the visual cortices of the monkey, neurons have been found to fire even before a saccade brings a stimulus into their receptive field (Nakamura and Colby 2002;Unema and Goldberg 1997). This suggests that this neural activity may correspond to a predictive
updating of the visual representation. This mechanism might explain, at least partially, why eye tracking of predictable targets such as one’s own hand can be per-formed with almost no latency (Miall and Reckess 2002; Vercher et al. 1991). Neurophysiological findings also indicate that large groups of neurons in the cerebellum display a preferential tuning to arm position and move-ment direction. These neurons also fire with almost no latency when compared to real arm dynamics (Fu et al. 1997;Roitman et al. 2005). Together, these results highly suggest that the brain may use internal forward models in order to predict the consequences of upcoming move-ments.
Based on these biological results and hypotheses, our model proposes a neural mechanism that links the veloc-ity tuning properties of groups of neurons to their abil-ities to update their own sensory representation through velocity integration. For instance, by using such a mech-anism, the overall integration time of gain fields would be reduced drastically and thus would result in more efficient sensorimotor transformations. Moreover, neu-rons having a preferential tuning to position and velocity like those of our model have been found in the cere-bellum (Roitman et al. 2005), a brain area suggested to be involved in both the inverse and forward control of movements (Miall and Reckess 2002; Vercher and Gauthier 1988;Wolpert and Kawato 1998). Therefore, in addition to the suggested role of neurons of the cer-ebellum and of motor cortices in the direct control of movements (Georgopoulos 1996;Schweigenhofer et al. 1998;Todorov 2000), their sensitivity to nonlinear mix-tures of information, such as arm position and velocity, may also suggest an intrinsic dynamic process within a cortical column. As proposed by our model architecture, their firing might be used directly within their internal representation for a rapid and predictive-like updating, which therefore would not need to wait for slow sensory feedbacks.
4.3 Shared representations: motor imagery and imitation
Our model also has some implications concerning motor imagery and imitation-related mechanisms. First, motor imagery is the ability to mentally imagine oneself or someone else performing a movement. Recent find-ings suggest that this mental operation is performed in motor terms, i.e., by activating parts of the motor cor-tices that would be effectively involved in overt move-ment execution (Fogassi and Gallese 2002;Jeannerod and Decety 1995;Porro et al. 1996). Computationally, this hypothesis implies that some motor areas, receiving projections from proprioceptive feedback during
nor-mal motor execution, should also be able to process motor commands without being influenced by this afore-mentioned information. Under this problematic issue, our model analysis provides some insight as to a poten-tial neural mechanism resolving this problem. Indeed, it suggests that the same neural substrate, by increas-ing its global excitation level, can detach itself from the efferent sensory inputs, and hence perform this imag-ery task freely. By lowering this excitation, the external inputs can lock the network back to the actual sensory state, resetting its internal representation. In addition, our model may also, to some extent, contribute to the explanation of the neural mechanisms underlying illu-sory perception of one’s own body resulting from epilep-tic seizures, such as the autoscopic phenomenon (Blanke and Mohr 2005;Blanke et al. 2002). Indeed, since epilep-tic seizures are defined as an abnormal synchronization of electrical neuronal activity, this would correspond in our model to an abnormally high excitation of the net-work. Let us consider, for example, the proprioceptive and the vestibular systems, which are responsible for the representation of self-body schema and self-localization in space. An epileptic seizure affecting these systems would produce in an “unlocking” or loss of control of the network representation from the real sensory state. As mentioned by the authors of these studies, such an effect may result in a disintegration of self processing in brain areas related to self and others (Blanke and Mohr 2005). Backpropagating this conflicting information to visual areas may produce the reported hallucinations.
Furthermore, shared representations have also been found when considering movement observation and exe-cution. Indeed, recent neurophysiological studies indi-cate that both the observation and the execution of actions activate a shared complex of brain areas, usually called the mirror neuron system (Iacoboni et al. 1999; Rizzolatti et al. 2001). The discovery of this network of brain regions has led to several very attractive hypothe-ses related to the underlying neural mechanisms of imi-tation and of theory of mind (Arbib 2002;Decety and Sommerville 2003;Gallese and Goldman 1988;Keysers and Perrett 2004;Wolpert et al. 2003). It is, for instance, suggested that when we observe another human being performing an action, we unconsciously simulate the same action with our own representation of our body and world. This may allow us, to some extent, to predict what the outcome of that observed movement will be. However, as with the problem of the shared represen-tation during motor imagery, this calls for explanations as to what happens when we simultaneously execute an action and observe others, and as to how the brain min-imizes conflicts arising from these potentially contra-dictory sources of information. Again, a partial answer
ers is present. Moreover, our model also predicts that errors may increase with the decreasing ability of the network to update its internal representation accurately with speed. At very high movement velocities, the pat-tern of errors should reflect a bias toward a boundary corresponding to the internal limit for movement inte-gration.
Finally, our modeling study also gives some insights on a plausible neural medium for learning by imita-tion through motor resonance (Gallese and Goldman 1988;Oztop et al. 2006;Wolpert et al. 2003). This princi-ple has been proposed to explain the mirror response of the neurons in premotor areas by suggesting that the perception of others’ movements and actions acti-vates in parallel our internal representations of motor plans. Through competition, the plan corresponding best would be selected and may further be used for predic-tion of movement outcome, acpredic-tion understanding, and imitation. Since our model can potentially represent any state space, it may implement a motor plan by setting its intrinsic dynamics to correspond to that of the motor plan. Then, the state of a demonstrator’s movement may drive the network. And finally, by monitoring the mean energy of the neural ensemble that is maximal when the internal dynamics match that of the external input, it may thus be possible for the brain to select the best among activated motor plans. Moreover, as already sug-gested by earlier modeling works (Demiris and Hayes 2002;Oztop et al. 2006;Wolpert et al. 2003), imitation processes may use such a comparison value given here by our energy function to perform a gradient ascent on that function. Within this framework, movement imi-tation would consist in maximizing the energy of the shared representation of self and others’ movements.
5 Conclusion
This paper describes an extension of classical dynamic neural field models, which are known to exhibit mar-ginal attractor states corresponding to compact activity packets on the neural surface. Our extension provides additional properties to these models. One such prop-erty is to allow the network to modify its intrinsic
nal stimuli and the network intrinsic dynamics. Finally, the biological plausibility of the model and its impli-cations concerning several brain computational mecha-nisms were discussed in light of neurophysiological and behavioral data. The discussed topics include velocity tuning to visual stimuli, sensory discrimination, senso-rimotor transformations, motor control, motor imagery, and imitation.
Acknowledgements E. L. Sauser would like to specially thank G. Schöner for a helpful discussion concerning the problem of response latency occurring in recurrent models of sensorimotor transformations. This work was supported by the Swiss National Science Foundation, through grant no. 620-066127 of the SNF Professorships program.
A A coupled attractor model
In this paper, the following continuous attractor network dynamics are considered;
τ ˙u(r, s, t) = −u(r, s, t) + h(s, t) + x(r, s, t) +‹ W(r − r) − λ∇W(r − r) · s
×fu(r, s, t)drds, (17) where u(r, s, t) is the membrane potential of a neuron with time constantτ, preferentially tuned to r, a vari-able in stimulus spaceDr, and to s ∈ Ds. We consider both a ring and a torus attractor (Table 1). f(u) is the activation function chosen to be the linear threshold function: max(0, u). W is a center-surround, symmetric, and Gaussian-like recurrent weights kernel, λ > 0 a scaling factor, and∇ the gradient operation along r. As will be shown later, a resulting effect of the second term of the recurrent connectivity is that the neuron’s sensi-tivity to the variable s implicitly corresponds to a pre-ferred movement direction along the other variable r. The external inputs are decomposed into two parts: the background input h(s, t) and the stimulus inputs x(r, s, t). In what follows, our interest is primarily in the dynam-ics of the interactions between the proposed network and its external inputs. These interactions will be shown to modify the attractor states of the network. We will
begin by introducing the case where the background input alone is sufficient to drive the network marginal attractor states to become marginal linear trajectories in the spaceDr. Then, we will consider how a stimulus input should be defined in order to drive the network toward its own motion dynamics. We will first define a nonlinear form of input, since its derivation from the background input form is the most straightforward. Then, we will slightly modify this input description by defining a lin-ear form of it, which is more suitable for transferring information across neural populations. Indeed, this lin-ear form of input will further help us to derive adequate synaptic projections so that, within a larger network of neural populations, instances of our model may transfer their encoded information across each other. Further, since such network dynamics can only be described fully analytically for certain rare special cases (Amari 1977; Sauser and Billard 2005;Xie et al. 2002;Zhang 1996), we will assume some linear approximations around equi-librium points. We start by considering general stable solutions as existing and then develop our mathemati-cal argument based on them. Indeed, the nature of the marginally stable solutions of such systems, known as activity bumps, has already been long described in the literature (Amari 1977;Salinas and Thier 2000;Sauser and Billard 2005;Wilson and Cowan 1973;Xie et al. 2002;Zhang 1996).
A.1 Static case
Before addressing inhomogeneous background inputs along s, let us begin by addressing the static case, where the background input is balanced and where no spa-tial input is applied to the network, i.e., respectively, h(s, t) = h0and x(r, s, t) = 0. Omitting the system vari-ables, we consider a marginally stable and static solution given by uand f= f (u). In this case, since the system is constant in s, the second term of the recurrent con-nectivity vanishes through the closed integral, which, by rewriting Eq. (17), gives
u− ˛
W∗ fds = h0, (18)
where ∗ denotes the convolution operation along r. Then, in order to extract the homogeneous term h0from the solutions, we define a normalization of these solu-tions by the following substitution:
u= h0U0 and f= h0F0, (19) where U0(r) and F0(r) are solutions only defined on r. Further, rewriting Eq. (18) gives
U0− 1 = ˛
W∗ F0ds. (20)
From this, the multiplicative effect of the background homogeneous input h0 on the solutions of the system can be observed. This property is the basis of the non-linear behavior of the neural fields in general (Salinas and Thier 2000;Sauser and Billard 2005). This implies that we can restrict our analysis, without loss of gener-ality, by considering normalized solutions of the system, since h0only acts as a scaling factor.
A.2 Velocity integration
We are now interested in how the background input may drive a stable activity packet toward a similar but travel-ing bump with velocity valong the space defined on r. A usual transformation performed in such a situation is to focus on a frame of reference moving with the bump so that, in this new frame, it appears static (Xie et al. 2002;Zhang 1996). We consider the following variable substitution: ˜r = r − t ˆ 0 v(t)dt. (21)
In this new frame,˙u(r, s, t) = −v(t)·∇u(˜r, s, t)+ ˙u(˜r, s, t), where ∇ is the gradient operator along ˜r. This leads Eq. (17) to become
− τv(t) · ∇u(˜r, s, t)
+τ ˙u(˜r, s, t) = −u(˜r, s, t) + h(s, t) + x(˜r, s, t) +‹ W(˜r − ˜r) − λ∇W(˜r − ˜r) · s
×fu(˜r, s, t)d˜rds. (22)
As mentioned earlier in the static case, a constant input h(s, t) = h0 leads the system to be constant along s, which suppresses the second term of the recurrent con-nectivity. Moreover, as can be guessed, this term will be responsible for compensating the new term on the left-hand side of Eq. (22). Therefore, we need to break the symmetry by adding ˆh0(t) · s, an asymmetric term, to the driving background input such that
h(s, t) = h0[1 + ˆh0(t) · s], (23)
where ˆh0corresponds to the normalized strength of the asymmetry breaking relative to the constant term h0. If, for a while, we omit terms containing a gradient expres-sion in Eq. (22), an approximate solution˜ualong s can be found, which is equivalent to Eq. (19) up to a constant given by the asymmetric term in Eq. (23). In a compact form, the solution of the system can be written as
0
whereγ0corresponds to the linear approximation factor, which, except for rare cases, may not be found analyti-cally. It corresponds to the slope of the relation between the asymmetry of the background input and the velocity of the network response. It strictly depends on the recur-rent weight profile, whose convolution is hard to solve analytically (Zhang 1996). Now, we return to the full form of Eq. (22) and show how the background input should be set in order for the network representation to exhibit the desired traveling activity profile. Let us begin by considering the recurrent synaptic drive. First, developing its symmetric part gives
˛ W∗ ˜fds (24b= h) 0 ˛ W∗ [1 + γ0ˆh0· s] F0ds = h0 ˛ W∗ F0ds+ γ0 ˛ W∗ F0[ˆh0· s]ds (20) = h0[(U0− 1) + 0] = h0(U0− 1).
Similarly, considering its asymmetric component, we obtain −˛ λ∇W · s∗ ˜fds (24b) = −h0λ ˛ ∇W · s∗ [1 + γ0ˆh0· s] F 0ds = −λh0 ˛ ∇W · s∗ Fds + γ0 ˛ ∇W · s∗ F0[ˆh0· s]ds = −λh0 0+ γ0ˆh0· ∇[W ∗ F0] (20) = −λγ0h0ˆh0· ∇(U0− 1) = −λγ0h0ˆh0· ∇U0.
And, finally, the complete recurrent synaptic drive is given by ˛ W− λ∇W · s∗ ˜fds= h0 U0− 1 − λγ0ˆh0· ∇U0 . (25) Then, by substituting Eqs. (23), (24a), and (25) into the system Eq. (22), we find that the velocity of the activity blob is determined by the asymmetric component of the background input, following
v(t) ≈ λ γ0
τ ˆh0(t) ⇔ ˆh0(t) ≈ τ λ γ0
v(t). (26)
ciently strong to completely inhibit one of the sublayers. However, in the torus case, because of the continuous nature of the space Ds spanned by the sublayers, one sublayer may eventually drive the network alone only for ˆh0 → ∞. As a consequence, regarding the net-work dynamics, the close integral along s of the recur-rent connectivity may be removed. The full recurrecur-rent synaptic drive thus becomes
W(˜r − ˜r) − λ∇W(˜r − ˜r) · s∗ ˜f(˜r, s, t). (27) where sdenotes the only active sublayer given by s = ˆh0/ˆh0. This recurrent drive has already been shown to lead a traveling activity peak to move with a con-stant velocity equal toλ/τ and with direction s(Zhang 1996). This is thus the integration limit that our model can reach.
B Stimulus input B.1 General input form
In this section, we are interested in how a stimulus input that conveys information related to the spatial localiza-tion of a stimulus may drive the network representalocaliza-tion to reflect the motion dynamics of that input. We con-sider a stimulus located at r0(t) in the stimulus space and moving in phase with the network intrinsic velocity v(t), i.e., ˙r0(t) = v(t). The shape G of the stimulus input is assumed to be a Gaussian-like shape centered on its cur-rent location r0(t). Moreover, in order to compensate for the neural dynamics and for the effect of the network recurrent connections, an additional differential term and a scaling factor are respectively needed to define a “well-behaving” input, i.e., one that does not turn into an asymmetric activity peak in the network representa-tion when moving within the neural field. It is defined by
x(r, s, t) = h1 Gr− r0(t) −τ ˙r0(t) · ∇G r− r0(t) 1+ ˙r0(t) · s , (28) where is a temporary constant. As can be noticed while looking at the network dynamics, the second differential term here compensates for the similar term in Eq. (22).
